Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $q = \dfrac{p - 10}{7p + 49} \times \dfrac{p^2 + 7p}{p^2 - 7p - 30} $
First factor the quadratic. $q = \dfrac{p - 10}{7p + 49} \times \dfrac{p^2 + 7p}{(p - 10)(p + 3)} $ Then factor out any other terms. $q = \dfrac{p - 10}{7(p + 7)} \times \dfrac{p(p + 7)}{(p - 10)(p + 3)} $ Then multiply the two numerators and multiply the two denominators. $q = \dfrac{ (p - 10) \times p(p + 7) } { 7(p + 7) \times (p - 10)(p + 3) } $ $q = \dfrac{ p(p - 10)(p + 7)}{ 7(p + 7)(p - 10)(p + 3)} $ Notice that $(p + 7)$ and $(p - 10)$ appear in both the numerator and denominator so we can cancel them. $q = \dfrac{ p\cancel{(p - 10)}(p + 7)}{ 7(p + 7)\cancel{(p - 10)}(p + 3)} $ We are dividing by $p - 10$ , so $p - 10 \neq 0$ Therefore, $p \neq 10$ $q = \dfrac{ p\cancel{(p - 10)}\cancel{(p + 7)}}{ 7\cancel{(p + 7)}\cancel{(p - 10)}(p + 3)} $ We are dividing by $p + 7$ , so $p + 7 \neq 0$ Therefore, $p \neq -7$ $q = \dfrac{p}{7(p + 3)} ; \space p \neq 10 ; \space p \neq -7 $